Class Notes (836,580)
Canada (509,856)
ACTSC 445 (24)
Lecture

unit 6 immunization

20 Pages
117 Views
Unlock Document

Department
Actuarial Science
Course
ACTSC 445
Professor
Jiahua Chen
Semester
Fall

Description
ACTSC 445: Asset-Liability Management Department of Statistics and Actuarial Science, University of Waterloo Unit 6 Immunization References (recommended readings): Chap. 3 of Financial Economics (on reserve at the library: call number HG174 .F496 1998). What is immunization? Redington (1952): Immunization implies the investment of assets in such a way that existing business is immune to a general change in the rate of interest. Fisher-Weil (1971): A portfolio of investment is immunized for a holding period if its value at the end of the holding period, regardless of the course of rates during the holding period, must be at least as it would have been had the interest rate function been constant throughout the holding period. Implication: If the realized return on an investment in bonds is sure to be at least as large as the appropriately computed yield to the horizon, then that investment is immunized. An immunization strategy is a risk management technique designed to ensure that for any small change in a specied parameter, a portfolio of debt instruments (e.g., T-bills, bonds, GICs etc) will cover a liability (or liabilities) coming due at a future date (or over a period in the future). It is a passive management technique because it takes prices as given and then tries to control the risk appropriately. (By contrast, active management techniques try to exploit changes in (1) the level of interest rates, (2) the shape of the yield curve (3) yield spreads, by using interest rate forecasts and identication of mispriced bonds) asset allocation problem (i.e., must choose assets that will produce an immunized portfolio) Single-liability case Well start with the case where there is only one liability in the portfolio, with corresponding cash ow of Ltat some time t. The goal is to choose an asset cash ow sequence {A ,t t 0} that will, along with L , protuce an immunized portfolio. Lets start with an example. Example I: Suppose an insurance company faces a liability obligation of $1 million in 5 years. The available market instruments are: 3-year, 5-year and 7-year zero-coupon bonds, each yielding 6% annual eective rate. Portfolio A: Invest $747,258.17 in the 5-year zero coupon bond 1 Portfolio B: Invest the same amount (i.e. $747,258.17) in a 3-year zero coupon bond. The maturity value at t = 3 is $889,996.44. Portfolio C: Invest $747,258.17 in a 7-year zero coupon bond. The maturity value at t = 7 is $1,123,600.00. If the yields remain unchanged, then the 3 portfolios have the same value of $1 000 000 at time 5. To verify if these portfolios are immunized or not, we need to look at what happens if, immediately after the portfolio is acquired, the yield changes instantaneously and remains constant at that level. First, note that for portfolio A, this change has no impact: its value at time 5 is still $1 000 000. But this is not true for portfolios B and C, as Tables 1 and 2 show. Table 1: Value of Portfolio B for dierent yields Value of Portfolio B Capital Gain Implied (%) at time 0 at time 5 at time 0 Yield (%) 4.00 791203.5944 962620.1495 43945.4215 5.20 5.00 768812.3874 981221.0751 21554.2146 5.60 5.90 749377.0511 998114.0975 2118.8782 5.96 6.00 747258.1729 1000000.0000 0.0000 6.00 6.10 745147.2753 1001887.6824 2110.8975 6.04 7.00 726502.2044 1018956.9242 20755.9684 6.40 8.00 706507.8685 1038091.8476 40750.3044 6.80 So for portfolio B, if the yields go up, then we realize a gain at time 5, because we can reinvest the proceeds obtained at time 3 at a high yield. But if the rates drop, then we realize a loss at time 5. The problem here is the reinvestment risk. Table 2: Value of Portfolio C for dierent yields Value of Portfolio C Capital Gain Implied (%) in year 0 at time 5 at time 0 Yield (%) 4.00 853843.6549 1038831.3609 106585.4820 6.81 5.00 798521.5425 1019138.3220 51263.3697 6.40 5.90 752211.5711 1001889.4658 4953.3982 6.04 6.00 747258.1729 1000000.0000 0.0000 6.00 6.10 742342.0181 998115.8742 4916.1548 5.96 7.00 699721.6100 981395.7551 47536.5629 5.60 8.00 655609.8081 963305.8985 91648.3647 5.21 The situation here is opposite from what we face with Portfolio B: if the rates drop, then we can sell the 7-year zero bond at a higher price at time 5, which results in a gain. But a yield increase produces a loss. The problem here is the interest rate or price risk. 2 Observations from Example I With a single liability, the best immunization strategy is the one for which the asset cash ow coincides with the liability cash ow When asset cash ows occur prior to (or after) the liability cash ow, the portfolio is subject to reinvestment risk (or market/interest rate/price risk). A valid question is: could we construct a portfolio containing cash ows occuring before and after the liability due date that could be immunized? Motivation: Any initial capital loss may be oset in time by greater returns from reinvestment. Similarly, any initial capital gain may be oset in time by lower returns from reinvestment. Does there exist an optimum trade-o? I.e., a way to construct a portfolio like this that maximizes (in some sense) the gain? The following example studies this idea. Example II: Consider Portfolio D, which consists in an investment of $373629.0864 in 3-year zero- coupon bonds, and $373629.0864 in 7-year zero-coupon bonds. Their maturity values are, respectively, 444,998.22 and 561,800.00. Note that the Macaulay duration of this portfolio is 5. If the yields remain unchanged, then at t = 5 we have 373,629.0864 2 (1.06) = 1000000. If the rates change, then we get the following results: Value of Portfolio D Capital Gain Implied (%) at time 0 at time 5 at time 0 Yield (%) 4.0 822523.6247 1000725.7552 75265.4518 6.01538 5.0 783666.9650 1000179.6986 36408.7921 6.00381 5.9 750794.3111 1000001.7817 3536.1382 6.00004 6.0 747258.1729 1000000.0000 0.0000 6.00000 6.1 743744.6467 1000001.7783 -3513.5262 6.00004 7.0 713111.9072 1000176.3396 -34146.2657 6.00374 8.0 681058.8383 1000698.8731 -66199.3346 6.01481 Hence with this portfolio, a gain is realized at time 5 for all alternative yi considered... Note that at time 0, there is a capital loss for portfolio D. More generally, we can look at the value of this portfolio at time t if the initial yield goes from 6%. That is, we can consider the value V = 444998.22(1 + y )(3t)+ 561800.00(1 + y)(7t) t for t = 1,...,10 and dierent s. 3 if rate drops y if rate rises t 4.00% 5.50% 5.90% 6.00% 6.10% 6.50% 8.00% 0 822524 765169 750794 747258 743745 729913 681059 1 855425 807253 795091 792094 789113 777358 735544 2 889642 851652 842002 839619 837249 827886 794387 3 925227 898493 891680 889996 888321 881698 857938 4 962236 947910 944289 943396 942509 939009 926573 5 1000726 1000045 1000002 1000000 1000002 1000044 1000699 6 1040755 1055047 1059002 1060000 1061002 1065047 1080755 7 1082385 1113075 1121483 1123600 1125723 1134275 1167215 8 1125680 1174294 1187650 1191016 1194392 1208003 1260592 9 1170708 1238880 1257722 1262477 1267250 1286523 1361440 10 1217536 1307018 1331927 1338226 1344552 1370147 1470355 Equivalently, we can look at the corresponding implied yield for the portfolio, which is the value i such that 747,258.17 = V (1 + i)t. t if rate drops y if rate rises t 4.00% 5.50% 5.90% 6.00% 6.10% 6.50% 8.00% 1 14.48 8.03 6.40 6.00 5.60 4.03 1.57 2 9.11 6.76 6.15 6.00 5.85 5.26 3.11 3 7.38 6.34 6.07 6.00 5.93 5.67 4.71 4 6.53 6.13 6.03 6.00 5.98 5.88 5.52 5 6.02 6.00 6.00 6.00 6.00 6.00 6.01 6 5.68 5.92 5.98 6.00 6.02 6.08 6.34 7 5.44 5.86 5.97 6.00 6.03 6.14 6.58 8 5.26 5.81 5.96 6.00 6.04
More Less

Related notes for ACTSC 445

Log In


OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Sign up

Join to view


OR

By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.


Submit